US 7110003 B2 Abstract The method
600 renders a self-overlapping polygon, wherein the polygon is a set of one or more closed curves each comprising line segments. The method performs, for a currently scanned pixel that overlaps both sides of a line segment of the self-overlapping polygon within a currently scanned scanline, the following steps. The method 600 decomposes 618 that portion of the polygon that lies within the currently scanned pixel into a number of closed loops comprising at least those portions of those line segments that lie within the currently scanned pixel, the closed loops are such that when they are combined the combination is substantially equivalent to that portion of the polygon that lies within the currently scanned pixel. The method 600 combines 623 incrementally the closed loops and determines one or more winding count values representative of respective weighted averages of winding counts of the combined closed loops. The method 600 then determines 622 a real opacity of the currently scanned pixel according to a predetermined fill rule utilizing an intrinsic opacity of said polygon and the one or more winding count values. The method 600 finally renders 624 the currently scanned pixel with the determined real opacity.Claims(21) 1. A method of rendering objects, the method comprising, for each object within a scanline, the steps of:
determining each boundary pixel that overlaps both sides of a border of the object;
computing a real opacity of each said boundary pixel, wherein the real opacity of a said boundary pixel is dependent upon the real opacities of subregions of said boundary pixel, and values representative of the areas of the respective subregions with respect to the total area of the boundary pixel, wherein the real opacity of each subregion is 1−(1−α)
^{|n| }where α is the intrinsic opacity of the object and n is the winding count for the subregion, and wherein the real opacity is a weighted sum of the real opacities of respective subregions, where the real opacities subregions are weighted with said values; andrendering each said boundary pixel by compositing using the corresponding computed real opacity.
2. A method as claimed in
computing the real opacities at a plurality of sampling points within each said boundary pixel; and
determining the real opacity of each said boundary pixel, wherein the real opacity of a said boundary pixel is the sum of the computed real opacities at the sampling points of said boundary pixel divided by the total number of sampling points in the pixel.
3. A method as claimed in
determining those subregions within each said boundary pixel which have a constant winding count;
computing the real opacities of a plurality of subregions within each said boundary pixel; and
determining the real opacity of each said boundary pixel, wherein the real opacity of a said boundary pixel is the sum of the product of percentage areas of pixel occupied by each subregion of said boundary pixel and its computed real opacity.
4. A method as claimed in
5. A method as claimed in
6. A method as claimed in
computing the real opacities for winding counts 1 to m, where m is a positive integer.
7. A method as claimed in
determining inner pixels of the object, the inner pixels being pixels inside the object other than said boundary pixels;
computing a real opacity of each said inner pixel, wherein the real opacity is dependent upon an intrinsic opacity of the object and a winding count for that inner pixel; and
rendering each inner pixel with the corresponding determined real opacity.
8. Apparatus for rendering objects, the apparatus comprising processing means for processing each object within a scanline, the processing means comprising:
means for determining each boundary pixel that overlaps both sides of a border of the object;
means for computing a real opacity of each said boundary pixel, wherein the real opacity of a said boundary pixel is dependent upon the real opacities of subregions of said boundary pixel, and values representative of the areas of the respective subregions with respect to the total area of the boundary pixel, wherein the real opacity of each subregion is 1−(1−α)
^{|n| }where α is the intrinsic opacity of the object and n is the winding count for the subregion, and wherein the real opacity is a weighted sum of the real opacities of respective subregions, where the real opacities subregions are weighted with said values; andmeans for rendering each said boundary pixel by compositing using the corresponding computed real opacity.
9. Apparatus as claimed in
means for computing the real opacities at a plurality of sampling points within each said boundary pixel, and
means for determining the real opacity of each said boundary pixel, wherein the real opacity of a said boundary pixel is the sum of the computed real opacities at the sampling points of said boundary pixel divided by the total number of sampling points in the pixel.
10. Apparatus as claimed in
means for determining those subregions within each said boundary pixel which have a constant winding count;
means for computing the real opacities of a plurality of subregions within each said boundary pixel; and
means for determining the real opacity of each said boundary pixel, wherein the real opacity of a said boundary pixel is the sum of the product of percentage areas of pixel occupied by each subregion of said boundary pixel and its computed real opacity.
11. Apparatus as claimed in
12. Apparatus as claimed in
13. Apparatus as claimed in
means for computing the real opacities for winding counts 1 to m, where m is a positive integer.
14. Apparatus as claimed in
means for determining inner pixels of the object, the inner pixels being pixels inside the object other than said boundary pixels;
means for computing a real opacity of each inner pixel, wherein the real opacity is dependent upon an intrinsic opacity of the object and a winding count for that inner pixel; and
means for rendering each inner pixel by compositing using the corresponding determined real opacity.
15. A computer program for rendering objects, the computer program comprising processing code for processing each object within a scanline, the processing code comprising:
code for determining each boundary pixel that overlaps both sides of a border of the object;
code for computing a real opacity of each said boundary pixel, wherein the real opacity of a said boundary pixel is dependent upon the real opacities of subregions of said boundary pixel, and values representative of the areas of the respective subregions with respect to the total area of the boundary pixel, wherein the real opacity of each subregion is 1−(1−α)
^{|n|}, where α is the intrinsic opacity of the object and n is the winding count for the subregion, and wherein the real opacity is a weighted sum of the real opacities of respective subregions, where the real opacities subregions are weighted with said values; andcode for rendering each said boundary pixel with by compositing using the corresponding computed real opacity.
16. A computer program as claimed in
code for computing the real opacities at a plurality of sampling points within each said boundary pixel, and
code for determining the real opacity of each said boundary pixel, wherein the real opacity of a said boundary pixel is the sum of the computed real opacities at the sampling points of said boundary pixel divided by the total number of sampling points in the pixel.
17. A computer program as claimed in
code for determining those subregions within each said boundary pixel which have a constant winding count;
code for computing the real opacities of a plurality of subregions within each said boundary pixel; and
code for determining the real opacity of each said boundary pixel, wherein the real opacity of a said boundary pixel is the sum of the product of percentage areas of pixel occupied by each subregion of said boundary pixel and its computed real opacity.
18. A computer program as claimed in
19. A computer program as claimed in
20. A computer program as claimed in
code for computing the real opacities for winding counts 1 to m, where m is a positive integer.
21. A computer program as claimed in
code for determining inner pixels of the object, the inner pixels being pixels inside the object other than said boundary pixels;
code for computing a real opacity of each inner pixel, wherein the real opacity is dependent upon an intrinsic opacity of the object and a winding count for that inner pixel; and
code for rendering each inner pixel with the corresponding determined real opacity.
Description The present invention relates generally to rendering objects, and in particular to rendering a self-overlapping polygon. Scan conversion is a computer graphics process in which geometrical objects such as lines and polygons are converted into pixel data for displaying on a raster device. Scan conversion can either operate in a simple “aliased” or more sophisticated “anti-aliased” mode. In aliased mode, each display pixel is assigned one of two values according to whether it is classified as being inside or outside of the object being scan converted. It is well known that in this mode, “aliasing” effects can occur along the edges of objects, which give rise to “jagged” appearance. Techniques aimed at reducing or eliminating this effect are referred to as anti-aliasing techniques. They work by blending the object's colour with the colour of the background along pixels lying on the edges of the object to create smooth transitions between pixels lying outside and inside the object. Scan conversion methods that incorporate these techniques are said to be operating in “anti-aliased” mode. Existing anti-aliasing techniques generally fall into one of three types: filtering, multi-point sampling, and area sampling. Filtering techniques work by applying a low-pass filter to the pixel values produced by an aliased scan conversion of an object to remove the high spatial frequency components that give rise to the “jagged” appearance of edges. Since this requires a matrix multiplication at each pixel, filtering techniques are computationally expensive. Filtering also has the effect of blurring horizontal and vertical edges that fall exactly on pixel boundaries that would otherwise appear sharp when scan converted using other anti-aliasing techniques, which may be undesirable. In multi-point sampling, each pixel is sampled at several different locations at which tests are made to identify which of these points lie inside the object being scan converted. The pixel is then assigned a value based on the number of such points. Variations of this technique also exist where sampling is performed along several continuous horizontal or vertical line segments within each pixel instead of at discrete locations. The pixel is assigned a value based on the total length of these line segments that lie inside the object being scan converted. A disadvantage of multi-point sampling is that it does not handle thin edges well. Consider the examples shown in A similar problem exists in The third type of anti-aliasing technique is area sampling, where each pixel is assigned a value according to the percentage area of the pixel that falls inside the object being drawn. Its advantage over multi-point sampling is that it does not suffer from the problem illustrated in When scan converting a complex or self-overlapping polygon, it is necessary to select a fill rule. A polygon is a set of one or more closed curves each comprising of a number of vertices connected by straight line segments. Each closed curve is also known as a contour and has an associated direction. A polygon is said to be simple if it comprises of only a single contour, otherwise it is said to be complex. A polygon is also said to be self-overlapping or self-intersecting if one or more of its contours crosses over itself or over other contours. Examples of the possible different types of polygons are shown in Odd-even and non-zero winding are two fill rules well known to those skilled in the art based on the winding count of a point. The winding count of a point is defined as follows: draw an arbitrary path from any point outside of the polygon to this point. Count the number of times a contour crosses the path from one side of the path to the other, and the number of times a contour crosses the path in the opposite direction. The winding count of the point is the obtained by subtracting the second number from the first. When using the odd-even fill rule, points that have odd winding counts are considered to be inside the polygon and are hence filled with the polygon's colour and opacity, whilst points with even winding counts are considered to be outside the polygon and are not filled. When using the non-zero winding fill rule, points that have zero winding counts are considered to be outside the polygon, whilst points with non-zero winding counts are considered to be inside the polygon. The publication U.S. Pat. No. 6,084,596 discloses a third fill rule, called “winding-counting”. Unlike the non-zero winding fill rules, in which all pixels classified as being inside the polygon are rendered with uniform colour and opacity, winding-counting assigns an opacity value to a pixel according to its absolute winding count. More specifically, pixels with a zero winding count are classified as being outside the polygon and are not filled, and pixels with a +1 or −1 winding count are filled in the same manner as in the odd-even and non-zero winding rule. The pixels with a winding count of n, where |n|>1, are rendered by performing |n| repeated rendering operations, each time using a pixel with the colour and opacity of the polygon. For fully opaque polygons, this produces identical results to the non-zero winding fill rule. For partially transparent, complex and/or self-overlapping polygons however, winding-counting gives the effect of the polygons being made up of several overlapping layers. A comparison between the three fill rules is shown in However, the scan conversion method of the publication U.S. Pat. No. 6,084,596 suffers from “aliasing” effects. It is an object of the present invention to substantially overcome, or at least ameliorate, one or more disadvantages of existing arrangements. According to one aspect of the invention, there is provided a method of rendering objects, the method comprising, for each said object within a scanline, the steps of: determining each boundary pixel that overlaps both sides of a border of the object; computing a real opacity of each said boundary pixel, wherein said real opacity of a said boundary pixel is dependent upon an intrinsic opacity of the object, winding counts for subregions of said boundary pixel, and values representative of the areas of the respective subregions with respect to the total area of the boundary pixel; and rendering each said boundary pixel by compositing using the corresponding computed real opacity. According to another aspect of the invention, there is provided apparatus for rendering objects, the apparatus comprising processing means for processing each said object within a scanline, the processing means comprising: means for determining each boundary pixel that overlaps both sides of a border of the object; means for computing a real opacity of each said boundary pixel, wherein said real opacity of a said boundary pixel is dependent upon an intrinsic opacity of the object, winding counts for subregions of said boundary pixel, and values representative of the areas of the respective subregions with respect to the total area of the boundary pixel; and means for rendering each said boundary pixel by compositing using the corresponding computed real opacity. According to still another aspect of the invention, there is provided a computer program for rendering objects, the computer program comprising processing code for processing each said object within a scanline, the processing code comprising: code for determining each boundary pixel that overlaps both sides of a border of the object; code for computing a real opacity of each said boundary pixel, wherein said real opacity of a said boundary pixel is dependent upon an intrinsic opacity of the object, winding counts for subregions of said boundary pixel, and values representative of the areas of the respective subregions with respect to the total area of the boundary pixel; and code for rendering each said boundary pixel by compositing using the corresponding computed real opacity. A number of preferred embodiments of the present invention will now be described with reference to the drawings, in which: Where reference is made in any one or more of the accompanying drawings to steps and/or features, which have the same reference numerals, those steps and/or features have for the purposes of this description the same function(s) or operation(s), unless the contrary intention appears. 1.0 Polygon Fill Rules Before proceeding with a description of the embodiments of the invention, a brief review on Polygon Fill Rules is discussed herein. A polygon is a set of one or more closed curves each comprising a number of vertices connected by straight-line segments. Each closed curve is also known as a contour and has an associated direction A polygon is said to be simple if it comprises only a single contour, otherwise it is said to be complex. A polygon is also said to be self-overlapping or self-intersecting if one or more of its contours crosses over itself or over other contours. Examples of the possible different types of polygons are shown in When scan converting a complex or self-overlapping polygon, it is necessary to select a “fill rule”. “Odd-even” and “non-zero winding” are two fill rules well known to those skilled in the art. When using the odd-even fill rule, points that have odd “winding counts” are considered to be inside the polygon and are hence filled with the polygon's colour and opacity, whilst points with even winding counts are considered to be outside the polygon and are not filled. When using the non-zero winding fill rule, points that have zero winding counts are considered to be outside the polygon, whilst points with non-zero winding counts are considered to be inside the polygon. The winding count of a point is defined as follows: draw an arbitrary path from any point outside of the polygon to this point. Count the number of times a contour crosses the path from one side of the path to the other, and the number of times a contour crosses the path in the opposite direction. The winding count of the point is the obtained by subtracting the second number from the first. A more formal definition of the odd-even fill rule is as follows; if the intrinsic opacity of an object at a point P is α, and P has a winding count of n, then point P is assigned a real opacity value of:
A more formal definition of the non-zero fill rule is as follows: if the intrinsic opacity of an object at a point P is α, and P has a winding count of n, then point P is assigned a real opacity value of
A third fill rule, called “winding-counting”, has also been proposed in U.S. Pat. No. 6,084,596 by George Politis (herein incorporated by reference). Unlike the non-zero winding fill rules, in which all pixels classified as being inside the polygon are rendered with uniform colour and opacity, the winding-counting fill rule assigns an opacity value to a pixel according to its absolute winding count. More specifically, pixels with a zero winding count are classified as being outside the polygon and are not filled, and pixels with a +1 or −1 winding count are filled in the same manner as in the odd-even and non-zero winding rule. Pixels with a winding count of n, where |n|>1, are rendered according to U.S. Pat. No. 6,084,596 by performing |n| repeated compositing operations, each time using a pixel with the colour and opacity of the polygon, For fully opaque polygons, this produces identical results to the non-zero winding fill rule. For partially transparent, complex and/or self-overlapping polygons however, the winding-counting fill rule gives the effect of the polygons being made up of several overlapping layers. A comparison between the three fill rules is shown in A short coming of the definition of the winding-counting fill rule given in U.S. Pat. No. 6,084,596 is that it does not address the issue of how polygons are rendered in anti-aliased mode. A difficulty lies in the treatment of pixel(s) that comprise subregions of different winding counts, as illustrated in The problem does not arise when operating in aliased mode since such pixels are always assigned a single winding count, based on certain criteria. For example, the pixel can be assigned the winding count of the point at the centre of the pixel, or the winding count of the largest region of the pixel. In any case, the pixel is then rendered by repeatedly performing a number of compositing operations equal to the absolute value of the assigned winding count. In anti-aliased mode however, it is necessary to take into account the different regions that make up the entire pixel. Thus it is not clear how many compositing operations are to be performed to render the pixel, nor it is clear what pixel value is to be composited each time. 1.2 New Definition of the Winding-counting Fill Rule The purpose of this section is to present a formal definition of a new winding-counting fill rule that will enable anti-aliased scan conversion methods to be used unambiguously. Consider the case where two simple, partially transparent polygons of opacity α are composited together using the “over” operator as defined in the publication Porter, Thomas, and Duff, “Compositing Digital Images” SIGGRAPH 84, 1984 (herein after referred to as Porter et. al). According to the definition of this operator, when a point with opacity of α In the present example, since the opacity of both polygons is α, the opacity of the intersection region between the two polygons is:
Now consider the case where there are n overlapping polygons instead of 2. Again each polygon is partially transparent with opacity of α. When these polygons are composited together using the “over” operator as defined in Porter et. al., it can be shown by mathematical induction that the resulting opacity of the intersection region between all n polygons is given by:
Since the aim of the winding-counting fill rule is to create the effect of polygons being made up of several layers in regions where the absolute winding counts are greater than 1, it is desirable that such regions are rendered with the same opacity as that would be obtained by compositing together the same number of such layers. The winding-counting fill rule is thus defined as follows: if the intrinsic opacity of an object at a point P is α, and P has a winding count of n, then point P is assigned a real opacity value of:
Under the winding-counting fill rule, objects are rendered according to their real opacities. In contrast, objects are rendered according to their intrinsic opacities under the non-zero winding fill rule. For pixels with integral winding counts, the above definition produces identical results to that given in U.S. Pat. No. 6,084,596, The new definition however, allows objects to be rendered more efficiently, since a pixel with a winding count of n needs to be composited only once rather than |n| times as described in U.S. Pat. No. 6,084,596. Although Eqn. (3) needs to be evaluated to determine the real opacity from the winding count of the pixel, its computational cost can usually be amortised over many pixels with the same winding counts and hence will likely be insignificant. An example of how this can be achieved is as follows: Given an object with uniform intrinsic opacity to be rendered, pre-compute the real opacities for points with absolute winding counts from 1 to m, where m is some positive integer. Store these in a look up table indexed by the absolute winding count. Then perform the scan conversion as described in U.S. Pat. No. 6,084,596, but instead of compositing a pixel whose winding count is n |n| times, do the following. If |n|≦m, then look up the pre-computed table created to obtain the real opacity associated with a winding count of |n|. Otherwise compute the real opacity using Eqn. (3). The above method takes advantage of the fact that for most cases, the maximum absolute winding count over the entire object is usually rather low, and hence the pre-computed look up table will likely cover the majority of pixels. 2A.0 First Arrangement A method The aforementioned new definition (Eqn. 3) allows existing anti-aliased scan conversion techniques to be extended to support the winding-counting fill rule. For example, in multi-point sampling methods, instead of rendering each pixel based on the total number of sampling points that lie inside the polygon, each pixel is rendered based on the sum of the real opacities at all sampling points. This is illustrated in The pixel is then assigned an opacity value equal to this sum divided by the total number of sampling points. Anti-aliased scan conversion methods based on area sampling can be extended to support the winding-counting fill rule in a similar way. Instead of rendering each pixel according to the percentage area of the pixel that falls inside the polygon, it is rendered according to the sum of the percentage areas of the different subregions that make up the pixel, weighted by their real opacities. This is illustrated in the example shown in where x,y are the percentage areas of the subregions whose winding count is 1, and z is the percentage area of the sub-region whose winding count is 2. Turning now to The method During step During step After completion of step The method If the decision block On the other hand, if the decision block After completion of either steps 2.0 Second Arrangement A method Turning now to The method During step After completion of step The steps During step The decision block In a further variation, the method On the other hand, if the decision block After completion of step The method In another variation of the method, the line segments are not approximated and area values defined by each contour segment are directly calculated. As will be apparent from the aforementioned sections, the size of any region contained within a pixel is expressed as the ratio of the area of the region within the pixel to the total pixel area, which total pixel area is taken as being one square unit. For the purposes of this specification, the term area value(s) is taken to mean such ratio(s), unless otherwise expressed or implied to the contrary. After completion of step Preferably, the method Also during this step The method in step The theory behind computing these winding count values and area values is described below in more detail in the sections “6.0 Weighted Average Winding Count Approximation”. “8.0 Merging a Constituent Region into an Accumulator” and “8.1 Merging Regions of the Clockwise and Counterclockwise Accumulators”. After completion of step After completion of steps For ease of explanation, the method Although the method The aforementioned rendering methods work by decomposes the coverage area of a polygon within each pixel into separate constituent regions, which constituent regions are simple regions whose winding counts and pixel coverage areas can be easily computed. The constituent regions are then combined together to obtain a weighted average of the winding counts and area values of the self-overlapping polygon in a relatively efficient and accurate manner. The method is preferably based on area sampling and is free from the defects of multi-point sampling associated with thin edges. It also overcomes the computational complexity suffered by conventional area sampling methods by computing approximate (rather than exact) area values. The rendering method also takes advantage of the fact that under most circumstances, the computation of the pixel coverage areas need not be very accurate at all. This is evidently shown by the existence of multi-point sampling anti-aliasing techniques. These techniques are essentially approximate forms of area sampling, where the pixel coverage areas are approximated by the number of sampling points that fall inside the polygon being scan converted. The defects of multi-point sampling techniques however highlight circumstances where accurate area computation is necessary—when scan converting thin lines. This is because the coverage area per pixel of thin lines is very small, and hence any small absolute variation in the computed area values would equate to a large percentage variation. If these variations occur over the length of a line, then it will appear to the viewer to have non-uniform thickness. As a result of the above observations, the rendering method has also been designed so that accurate area calculations are made when encountering pixels intersected by lines and similarly simple objects, whilst not affording the same level of accuracy to pixels intersected by more complicated edge combinations. This allows the rendering methods to achieve a similar level of efficiency to multi-point sampling techniques, but without their inherent defects. 3.0 Polygon Decomposition Turning now to For the purposes of this description, each length of the polygon from the point where the line segment of the polygon enters to the point where it leaves the pixel is referred to as a contour segment. In the above example, there are three contour segments a, b and c. The winding counts, for any given polygon, at all points within a particular pixel (and hence how the polygon is filled) is completely determined by the set of contour segments that reside fully inside the pixel boundary, and the initial winding count at a single reference point on the pixel boundary. Outside of the pixel boundary, the behaviour of the polygon is irrelevant as far as the pixel is concerned. This means that the interconnections between contour segments outside of the pixel can be rearranged ill any manner without affecting the final outcome of the winding counts within the pixel, as long as it does not alter the winding count of the reference point. The method Turning now to Depending on the initial winding count of the given reference point, and on how the reconnection of contour segments is performed, it may be necessary to introduce additional circular loops that enclose the whole pixel to ensure that the winding count of the reference point remains unchanged. For example, if the loop labelled A in the above Although the two polygons shown are theoretically equivalent, in practice they may give rise to different pixel coverage values due to the use of approximation techniques to be described later. Since approximations may be introduced whenever constituent regions are combined, polygons comprising fewer loops are usually more desirable as this generally implies fewer chances of errors. As a result, it is preferable to combine certain contour segments together to reduce the number of loops and hence further improve accuracy. Naturally, overly complicated loops whose coverage areas are difficult to compute should be avoided since they would defeat the purpose of polygon decomposition. Turning now to Having decomposed the polygon into a number of closed loops having constituent regions, the winding count of any point inside the pixel can simply be determined by counting the number of clockwise loops that enclose it, and subtracting the number of enclosing counter-clockwise loops (or vice-versa since the sign of the winding count is not important). Note that loops that are self-intersecting may enclose parts of the pixel in one direction, whilst enclosing other parts in the opposite direction. For example, the upper part of loop E in If two constituent regions are then merged, what happens at the intersection area is dependent on the relative directions of the enclosing closed loops. If the constituent regions are enclosed by loops of the same direction, then the absolute winding count of the intersection region increases, whereas if the constituent regions are of opposite directions, then the absolute winding count decreases. In the former, the two constituent regions are said to combine constructively, whereas in the latter, they are said to combine destructively. The situation is slightly more complicated when one or both constituent regions are self-intersecting, in which case parts of the intersection area may combine constructively, whilst other parts may combine destructively. For example, in From the areas of the two constituent regions, and the areas of the different intersecting subregions, it is possible to compute the overall opacity of the pixel. In the example of Under the non-zero winding fill rule, the opacity of the constructively combined intersecting region x is α instead of (2α−α Under the odd-even fill rule on the other hand, the opacity of the region denoted by x is 0, and hence the overall pixel opacity becomes
Although it is theoretically possible to derive similar equations for computing the opacity values of pixels comprising of three or more constituent regions, the resulting complexity would make it impractical. The method 4.0 Straight Line Approximation of Contour Segments Since computing the areas of regions defined by arbitrary contour segments is time consuming, the method Turning now to In an alternative approximation scheme, each contour segment is replaced by a single straight-line segment joining the entry and exit points, without passing through an intervening third point as in the preceding scheme. The scheme is illustrated in The latter scheme produces a faster method since area computations are simpler, but at the expense of degradation in accuracy. Whilst the loss of accuracy should not be significant under most situations, it can lead to undesirable artefacts where a contour segment enters and leaves on the same side of a pixel. Replacing such a contour segment with a single straight line would effectively remove it completely. This may produce 1-pixel gaps at the joint of connected edges. Preferably, the method To avoid the above problem without significantly sacrificing speed, a mixture of both schemes can also be used. For example, the earlier scheme can be used for approximating contour segments that enter and leave on the same side of a pixel, whereas the simpler scheme can be used for all remaining contours. The method For the purposes of this description, unless otherwise stated, the term “contour segment” will be used to refer to the straight-line approximation of a contour segment, rather than the contour segment itself. 5.0 Coverage Area of a Region Defined by a Single Contour Segment Turning now to The method With the preferred straight-line approximation scheme, the area of the region R defined by a single contour segment can be easily computed. For convenience and clarity, the list of possible cases has been divided into 4 sets. The formulae shown assume that origin is located at the top-left corner of the pixel, and that the x and y axes extend to the right and downwards respectively and have values between 0 to 1. 6.0 Weighted Average Winding Count As mentioned in Section “3.0 Polygon Decomposition”, that given a polygon comprising a number of constituent regions, the winding count at any point can be determined by computing the difference between the number of clockwise and counter-clockwise loops that enclose it. As also mentioned, the method Ideally, the accumulator needs to maintain the winding counts of all points inside the pixel, which has been found to be computationally intensive. It is thus preferable that the method This approximation scheme requires the storage of just two numbers in the accumulator: the area of the latter sub-region (herein referred to as the coverage area) and its weighted average winding count. It is actually more convenient to store the product of the area and the weighted average winding count, called the percentage winding count, rather than the weighted average winding count itself. From these parameters, overall opacity of the pixel can be determined. A drawback of the approximation scheme is that poor results are obtained when constituent regions whose winding counts have opposite signs are combined. As an example, reference is made to The above problem is remedied in the method 7.0 Combining Regions As mentioned previously, the method The preferred techniques for combining the aforementioned regions in either a constructive or destructive manner are described in this section 7.0 and the next sections 7.1 to 7.2. When two regions are to be combined constructively, the coverage area of the resulting region can be determined using the formula:
Similarly, when two regions combine destructively, the coverage area of the resulting region is given by
Unless it is known that the two regions are mutually exclusive, or that one region is subsumed by the other, computing their exact intersection area is computationally expensive. This is true even when contour segments have been approximated by straight lines as described in Section 4.0 To overcome this problem, the method It is worth noting that if x and y are within the range [0, 1], then the coverage areas produced by both Eqns (4) and (5) are also within the range [0, 1]. Although the above approximation works adequately well in most cases, it is inaccurate in situations where the two regions involved do not or hardly intersect one another, or where one region fully or almost subsumes the other. In the former, the true intersection area is close to zero, whereas in the latter it is close to either x or y. Experimental results suggest that the errors arising from the use of Eqn. (4) are sufficiently small to be not significantly noticeable. This is not necessarily the case for Eqn. (5) however. Due to the presence of the factor of 2, errors that arise from the approximation are effectively doubled, which can be significant. To overcome the problem, it is necessary to maintain geometrical information about each region in addition to its coverage area. Since execution speed is a priority, the method An advantage of this method is that it requires only very crude geometrical information about each constituent region. Preferably, this geometrical information concerning each constituent region is maintained in the form of a binary mask. 7.1 Mask Creation As mentioned earlier, the method preferably maintains geometrical information concerning each region defined by a contour segment. The geometrical information can be in the form of a binary mask where each bit indicates whether a certain sample point within the given pixel is inside or outside of the region that the mask represents. The sampling points in the mask need not follow the regular grid shown in The binary mask associated with each region defined by a single contour segment can be created by means of look up tables. In the simpler scheme where sampling points reside on the pixel boundary only, the mask is dependent only on the locations where the contour segment enters and leaves the pixel boundary. In other schemes that contain sampling points inside the pixel boundary, the mask is dependent not only on the locations of the entry and exit points, but also on the x and y coordinates of the intermediate vertex (if any) of the contour segment. Procedures for creating the necessary look up tables are not described since they should be familiar to those skilled in the art. 7.2 Mask Operations When two regions are combined, the binary mask of the resulting region can be obtained by performing an appropriate logical operation between the masks of the individual regions. If the regions combine constructively, then a logical OR operation is performed. If they combine destructively, then the resulting mask can be computed as the exclusive-OR of the two individual masks. However in the method The coverage area of each subregion are also easily computed using the formulae
To determine whether two constituent regions intersect, and if so, whether one region is subsumed by the other, a logical AND operation is performed between the masks of the individual regions. If the result is zero, then the two regions do not intersect. If the result is identical to one of the original mask, then the corresponding region is subsumed by the other. 8.0 Merging a Constituent Region into the Accumulators This section describes the procedure for merging a constituent region into a corresponding one of the two accumulators. It assumes that the constituent region being merged has a winding count whose absolute value is 1, and whose sign is identical to that of the accumulator in which it is to be merged. Consequently, the two regions are combined constructively. Let α, m Let z denote the area of the intersection region between the accumulator and the constituent region. As a result of the merge, the pixel may be divided into 3 subregions comprising of a subregion of area α−z and winding count p The resulting percentage winding count is then given by: As mentioned previously, the method Computation of the coverage area of the combined region is dependent on how the two regions intersect one another, which is in turn dictated by the interaction between the two binary masks m
The new coverage area denoted by α′ then replaces the original coverage area of the accumulator:
In a simpler variation of the present method, the intersection area between the constituent region and the accumulator is approximated by the product Aα, regardless of the values of m Finally, the binary mask of the accumulator is updated according to
As mentioned above, it is possible that two or more regions of the closed loops having the same winding count sign will intersect. As distinct from the winding counts, the area of any intersecting region contributes only once to the area value of the coverage area, as this area value is strictly representative of the area of the closed loops. 8.1 Merging Regions of the Clockwise and Counterclockwise Accumulators After all the clockwise and counterclockwise single closed loops have been merged into the respective clockwise and counterclockwise accumulators, the method The method The coverage area of each of these subregions is determined by the coverage areas currently, denoted by α
Namely, the clockwise accumulator is designated as storing a weighted average winding count w 9.0 Computing the Real Opacity of the Pixel The opacity of each of these subregions s One possibility for the winding-counting fill rule is to extend Eqn. (3) which relates absolute winding counts to real opacities to accept non-integral winding counts. The drawback of this approach is that it does not produce satisfactory results when the intrinsic opacity of the polygon is one (i.e. the polygon is fully opaque). In these cases, Eqn. (3) returns a value of 1 regardless of how small the absolute winding count is. The method Under the non-zero winding fill rule, opacity values are preferably capped at the intrinsic opacity, thus giving rise to the following equation: The winding count under the odd-even fill rule on the other hand, is “bounced” back and forth between the maximum and minimum values of 1 and 0 respectively. That is, if └n┘ is even, then the “effective winding count” is given by n−└n┘. When └n┘ is odd, it is 1−(n−└n┘). The opacity value is then computed as the product of the effective winding count and the intrinsic opacity, giving rise to the equation:
Notice again that the above equation produces the expected opacity value when n is a pure integer number. As described in section 8.1, merging the contents of the clockwise and counterclockwise accumulators can create upto three sub-regions of different winding counts, namely w In the first method, the overall real opacity of the currently scanned pixel is computed as the weighted average of the real opacities of the different sub-regions, weighted by their coverage areas. This overall real opacity is obtained using the following formulae:
In an alternative method, the overall opacity of the pixel is computed by computing first the average absolute winding count of the three sub-regions as well as the empty sub-region of the pixel, and then converting it into a overall real opacity value. This overall real opacity is obtained using the following formulae;
where f( ) is any one of the opacity functions specified in Eqns. (6), (7) or (8). In another alternative method, the overall opacity of the pixel is computed by first computing the average absolute winding count of the three non-empty subregions and then converting it into a real opacity value. In this case, the overall real opacity value is obtained using the following formulae:
After the overall opacity of the pixel has been computed, the method 10.0 The Rendering Method This section describes in more detail the steps As stated in Section 3 “Polygon Decomposition”, the winding counts at all points within a given pixel is fully determined by the set of contour segments appearing in the pixel, and the winding count of a single reference point on the pixel boundary (assuming that all contours that reside completely within the pixel have been removed by the straight line approximation scheme described in Section 4.0). Let this reference point be denoted by S, and let its winding count be W. Also, without loss of generality, let the winding counts be measured such that points enclosed by clockwise loops are given positive winding counts and points enclosed by counter-clockwise loops are given negative winding counts. Turning now to After completion of step For ease of explanation, the method The method After the method The decision block After completion of steps On the other hand, if the decision block The method As mentioned earlier, if the decision block During decision block A positive non-zero value of w indicates that, in addition to the contour segments that have been processed and whose resulting constituent region, have been added to the accumulators, the currently scanned pixel is also fully enclosed by |w| clockwise loops. The contribution of these loops will need to be added to the clockwise accumulator to obtain the true average winding count of the pixel. The method during step On the other hand if the decision blocks A negative non-zero value of w indicates that, in addition to the contour segments that have been processed and whose resulting constituent regions have been added to the accumulators, the currently scanned pixel is also fully enclosed by |w| counterclockwise loops. The contribution of these loops will need to be added to the counterclockwise accumulator to obtain the true average winding count of the pixel. The method during step On the other hand if the decision blocks During step Turning now to The decision block The decision block The method On the other hand, if the decision block Returning now to decision block The method The decision block Turning now to The method The method The method during this step Specifically, after completion of step The method during this step After completion of step On the other hand, if the decision block The method during this step After completion of step As mentioned previously, the steps If on the other hand, decision block After the completion of step The steps Turning now to If on the other hand, the decision block After the completion of either one of the steps Turning now to After completion of any of the steps After completion of step The method Although the method 10.1 Illustrative Example of the Preferred Rendering Method The method Tracing begins at point S, moving along the pixel boundary in the clockwise direction. The first contour segment encountered is b. Since this is the entry point of the contour segment, the winding count is decremented to −1. As this represents an increase in the absolute value of the winding count, tracing continues. The next contour segment that is encountered is a. Again this is an entry point and hence the winding count is decremented to −2. Tracing continues until the next contour segment c is reached. This time it is the exit point and hence the winding count is incremented back to −1. Since this represents a decrease in the absolute winding count, the current contour segment c is paired with the previous contour segment, a, to create a single constituent region. This region is formed by destructively combing the regions defined by the individual contour segments, as shown in The areas of regions A and C are then computed using the Type 2 formula shown in Following updating the accumulators, both contour segments a and c are removed from the pixel, leaving a single contour segment b remaining, as shown in Since b is also the only contour segment that remains, the constituent region is formed solely from b. The region formed includes points from P The contour segment b is then removed and tracing continues all the way back to point S, encountering no farther contour segments. The final winding count at S is zero, and hence no further updates are made to the two accumulators. Finally, the two accumulators are combined. Since the coverage areas of the accumulators do not intersect, as illustrated in Computing now the overall real opacity of the example pixel according to the winding-counting fill rule and using the afore-mentioned Eqns (6) and Eqns (9), the overall real opacity is:
On the other hand, the overall real opacity of the example pixel according to the non-zero winding fill rule and using Eqns (7) and (9) is:
Whilst, the overall real opacity of the example pixel according to tie odd-even ill rule and using Eqns (8) and (9) is:
Although the method Under the odd-even fill rule, when two single closed loops are combined, regardless of whether their constituent regions are defined by edges of the same or opposite directions, the end result in the intersection area is always destructive. In addition, the resulting coverage area can effectively be treated as if it has a winding count of 1. This means (i) it is no longer necessary to maintain a weighted average winding count as constituent regions are combined together, and (ii) only a single accumulator rather than two needs to be maintained for each pixel, since it is not necessary to distinguish between clockwise and counter-clockwise regions. When all constituent regions have been processed, the overall opacity of the pixel is then preferably the product of the final coverage area of the accumulator and the intrinsic opacity of the polygon. For example, when two single closed loops of the same direction are combined under the odd-even fill rule, the winding count of the intersection area will be even and thus makes no contribution to the real opacity of the pixel. Consequently, two single closed loops of the same direction can be combined destructively. When two single closed loops of opposing direction are combined under the odd-even fill rule, the winding count of the intersection area will be zero and also makes no contribution to the real opacity of the pixel. Consequently, two single closed loops of opposing direction can also be combined destructively. Thus when two constituent regions are combined, these can be combined destructively regardless of their direction. The remaining non-intersection areas have odd winding counts, and according to the odd-even fill rule these areas can effectively be treated as having a winding count of one. Thus under the odd-even fill rule, the weighted average of winding counts is effectively equal to the area value of the destructively combined regions. The real opacity of the currently scanned pixel can then be computed as being representative of the product of the intrinsic opacity of the polygon and the area value of the combined constituent regions. Turning now to After completion of step The method After the method The method The decision block The decision block The method The method The method The method The method Turning now to The method The method The method during this step Specifically, after completion of step The method On the other hand, if the decision block The method If on the other hand, decision block Turing now to After the completion of any one of the steps After completion of the updating step Returning to During decision block Opacity=aα, where a is the coverage area value currently stored in the accumulator and α is the intrinsic opacity of the polygon. In this particular case, as w is even the final winding count w makes no contribution to the overall real opacity of the currently scanned pixel. On the other hand, if the decision block Opacity=(1−a)α, where a is the coverage area value currently stored in the accumulator and α is the intrinsic opacity of the polygon. In this particular case, w is odd and the effect of adding to the accumulator an odd number of loops that fully enclose the pixel is equivalent to inverting the coverage area of the pixel. This is because the additional loops combine destructively with the accumulator. After completion of steps The aforementioned method(s) comprise a particular control flow. There are many other variants of the preferred method(s) which use different control flows without departing the spirit or scope of the invention. Furthermore one or more of the steps of the preferred method(s) may be performed in parallel rather sequential. 12.0 Preferred Apparatus The method(s) of The computer system The computer module Typically, the application program of the preferred embodiment is resident on the hard disk drive The method(s) of It is apparent from the above that the embodiments of the invention are applicable to the computer graphics and related industries. The foregoing describes only one some embodiments of the present invention, and modifications and/or changes call be made thereto without departing from the scope and spirit of the invention, the embodiment(s) being illustrative and not restrictive. Patent Citations
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